PROBLEM SET
Practice Problems for Exam #1
Math 1352, Fall 2004
Oct. 1, 2004
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This problem set has 9 problems.
Good luck!
Problem 1. Let R be the region bounded by the curves x = y 2 and y = x.
A. Find the volume of the solid generated by revolving the region R around the
x-axis.
B. Find the volume of the solid generated by revolving the region R around the
y-axis.
v
Problem 2.
The base of a solid is the region in the xy-plane bounded by the lines y = 2x
and x = 1. The cross sections of the solid perpendicular to the y-axis are
squares. Find the volume of the solid.
Problem 3. Find the area of one leaf of the four-leaf rose whose polar equation is r = sin(2θ).
Problem 4.
Find the area of the region that is inside the cardioid r = 1 + cos(θ) and
outside the circle r = 1.
Problem 5.
Find the arc length of the graph of
f (x) =
1 3 1 −1
x + x
3
4
on the interval from [1.2]. (This is Problem 9 on page 393 of the book.)
1
Problem 6. A tank has the shape of a hemisphere (see picture) of radius
1 meter. The tank is full of water, which weights 9800 N/m3 .
How much work is required to pump all of the water to a point one meter above
the top of the tank?
1
0.8
0.6
0.4
0.2
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
Problem 7.
The vertical cross sections of a tank are isosceles triangles, point upwards.
The bottom of the tank is 4 feet across and it is 8 feet high. If the tank is filled
with water to a depth of 4 feet, what is the total force on one end of the tank?
(The weight density of water is w = 62.4 lbs/ft3 .)
Problem 8. Let R be the region bounded by the curves y = 1 − x2 and
y = 1 − x.
A. Find the area of R.
B. Find the y-coordinate of the centroid of R.
C. Use the Theorem of Pappus to find the volume of the solid generated when
the region R is revolved around the x-axis.
2
Problem 9.
Find the following integrals.
A.
Z
x cos(2x) dx.
B.
Z
x2 ln(x) dx.
3